A necessary condition for local asymptotic stability of discrete-time nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in the control systems literature. In this paper, we investigate the problem of asymptotic stability for discrete-time nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Sundarapandian [1] for discrete-time nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Lin and Byrnes [2] for discrete-time nonlinear autonomous systems. We illustrate our result with several examples.